We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

中文翻译：

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大多数排列幂到一个小素数长度的循环

我们证明了大多数程度的排列$n$有一些功率，大约是一个素数长度的循环$\log n$. 明确地，我们证明对于$n$足够大，这些元素的比例至少是$1-5/\log \log n$与之间的素数$\log n$和$(\log n)^{\log \log n}$. 具有此性质的偶数排列的比例至少为$1-7/\log \log n$.